The Astrophysical Journal Letters, 792:L27 (4pp), 2014 September 10
C 2014.
doi:10.1088/2041-8205/792/2/L27
The American Astronomical Society. All rights reserved. Printed in the U.S.A.
OVERCOMING THE METER BARRIER AND THE FORMATION OF SYSTEMS
WITH TIGHTLY PACKED INNER PLANETS (STIPs)
1
A. C. Boley1 , M. A. Morris2 , and E. B. Ford3,4
Department of Physics and Astronomy, The University of British Columbia, 6224 Agricultural Road, Vancouver, B.C. V6T 1Z1, Canada
2 Center for Meteorite Studies, Arizona State University, P.O. Box 876004, Tempe, AZ 88287-6004, USA
3 Center for Exoplanets and Habitable Worlds, The Pennsylvania State University, 525 Davey Laboratory, University Park, PA 16802, USA
4 Department of Astronomy and Astrophysics, The Pennsylvania State University, 525 Davey Laboratory, University Park, PA 16802, USA
Received 2014 June 26; accepted 2014 July 23; published 2014 August 25
ABSTRACT
We present a solution to the long outstanding meter barrier problem in planet formation theory. As solids spiral
inward due to aerodynamic drag, they will enter disk regions that are characterized by high temperatures, densities,
and pressures. High partial pressures of rock vapor can suppress solid evaporation, and promote collisions between
partially molten solids, allowing rapid growth. This process should be ubiquitous in planet-forming disks, which
may be evidenced by the abundant class of Systems with Tightly packed Inner Planets discovered by the NASA
Kepler Mission.
Key words: minor planets, asteroids: general – planets and satellites: formation – protoplanetary disks
rapid migration of rocks and boulder-sized solids (∼10–100 cm)
in a nearly Keplerian disk (Adachi et al. 1976; Weidenschilling
1977). Because the disk has a pressure gradient, a monotonically decreasing pressure will cause the azimuthal speed of the
gas to always be less than the Keplerian orbital speed vK . A
solid, which does not have pressure support and moves at vK ,
will thus orbit with a head wind. This causes an exchange of
angular momentum, and the solid spirals inward. For very large
particle sizes, the stopping time (ts ) is very long compared with
the orbital period at 1 AU, preventing rapid in-spiral. For very
small particles, ts is very short, but the terminal radial velocity
is also very small, again preventing rapid in-spiral. Whenever
the term ts vK / = ts ΩK = τ ∼ 1, for disk radial distance and Keplerian orbital frequency ΩK , very efficient inward drift
occurs. Taking rough values for = 1 AU in an envisaged
planet-forming disk (ρ ∼ 10−9 g cc−1 and T ∼ 300 K), the
most rapid drift size corresponds to about 1 m, although notable
drift will begin in the millimeter to centimeter-size regime. For
pressure gradients in typical disk models, the in-spiral time for
a meter-sized object at 1 AU is only a few 100 yr, much shorter
than the timescale to form a planet at this location (see, e.g.,
Weidenschilling 1977). This appears to inhibit planet formation
and has been named the “meter barrier problem.” At both much
smaller and larger sizes, the in-spiral time becomes long relative
to 1 m.
An additional component of the meter barrier problem is
that high relative particle speeds, due to inward drift and/or
turbulence, will be destructive as sizes approach ∼10–100 cm.
This is emphasized in (Blum & Wurm 2008, see their Figure 12).
Even if large solids could be preserved at their given location
by turbulence, growth beyond about 10 cm becomes inhibited.
Previously proposed solutions include gravitational collapse of
the solids due to concentrations in, for example, streaming
instabilities (Youdin & Goodman 2005; Johansen et al. 2009;
Bai & Stone 2010).
We are thus left with two basic solutions to the meter barrier
problem: either (1) rapid formation of planetesimals must occur
through secondary instabilities, or (2) the collisional process and
inherent outcome of large solid interactions must be modified.
The work here explores the second option, which we argue
represents a fundamental process in disk evolution and planet
1. INTRODUCTION
The Kepler space Mission has revealed numerous planetary
types and systems, shaping our understanding of planet formation (Borucki et al. 2011; Batalha et al. 2013). Among the
quickly growing data is a subclass of multi-planet configurations referred to as Systems with Tightly packed Inner Planets
(STIPs). Their large abundance (>10% of stars) suggests that
they are one of the principal outcomes of planet formation.
The prototype STIP is Kepler-11 (Lissauer et al. 2011, 2013),
which hosts six known transiting planets, five of which have
measured masses in the super-Earth and mini-Neptune regimes.
The known planetary orbits in this system are spaced between
a = 0.09 and 0.47 AU, with small eccentricities and mutual
inclinations.
This dynamically cold configuration suggests that gravitational interactions between the planets were minimal during
formation or that the disk was strongly dissipative. The orbits
are not in low-order mean-motion resonances, a key signature
of smooth disk migration, suggesting that migration may not
have played a dominant role. Although disk turbulence could
be responsible for producing some STIPs that have planets near
commensurabilities (Pierens et al. 2011), in situ formation seems
to be the simplest solution for most STIPs (for a summary see
Raymond et al. 2008). Such formation would require growth
of massive planets on the stellar side of the water ice line,
which is difficult to reconcile with the current planet formation paradigm. Nonetheless, we must entertain the idea that the
current paradigm is incomplete and that super-Earth and miniNeptune formation at short orbital periods is plausible (Chiang
& Laughlin 2013). This requires the delivery and retention of
significant material into the inner nebula.
In this Letter, we demonstrate an overlooked mechanism that
should be prevalent in many planet-forming disks, lead to planet
formation at high disk temperatures, and overcome the meter
barrier.
1.1. The Meter Barrier
The “meter barrier” is the difficulty in gradually forming planetesimals from small solids, as aerodynamic forces will cause
1
The Astrophysical Journal Letters, 792:L27 (4pp), 2014 September 10
Boley, Morris, & Ford
100
PMg at 1X
PMg at 10X
PMg at 30X
PMg at 100X
Ps
Pressure (mbar)
10-1
10-2
10-3
-4
0.0 4
0.06
0.0 8
0.1
0.12
1250 K
1470 K
1600 K
10-5
1800 K
10
0.14
0.16
0.18
0.2
Radial Distance (AU)
Figure 1. Saturation vapor pressure of Mg (solid curve) compared with different hypothetical partial pressures (dashed/dotted curves). The solar volume mixing ratio
for Mg is taken to be 3.8 × 10−5 . Higher or lower values could be due to differences in metallicity or degree of concentration/depletion of solids. When the partial
pressure is above the saturation pressure, net evaporation becomes suppressed; large volume mixing ratio enhancements can suppress net evaporation at 1500 K.
Locally enhanced partial pressures due to self-shielding effects (vapor clouds) could also lead to equilibration.
formation. We further stress that the model presented here is
not intended to operate instead of, e.g., a streaming instability
(option 1), but represents a different pathway that may even
promote instabilities.
of forsterite (Mg2 SiO4 ), an abundant mineral in chondrites.
Following Richter et al. (2002), the evaporation rate is
2. SUPPRESSION OF ROCK EVAPORATION IN THE
INNER REGIONS OF PLANET-FORMING DISKS
The constants are set to match laboratory experiments at different temperatures and H2 pressures PH2 , which we calibrate here
using the figures and results presented in Richter et al. (2002)
for Mg. Thus, E0 = 300 kJ mole−1 and J0 = 75 mole cm−2 s−1
with P0 = 1 mbar. The evaporation coefficient is approximated
by γ ∼ 150 exp (−15000 K/T ).
Setting PH2 ∼ 10 mbar, T ∼ 1500 K, and the volume
mixing ratio r ∼ 3.8 × 10−5 (solar; Lodders 2003),5 we find
PMg /Ps ∼ 0.1 if the local Mg abundance is entirely vapor.
The implication is that in a solar composition nebula, a solid
enhancement greater than about 10 relative to bulk composition
will prevent rocks from evaporating for the given temperature
and pressure.
Consider a simple disk model that has a radial temperature
profile characterized by T ( ) = 300( /AU)−3/4 K and total
gas-mass density ρ( ) = 10−9 ( /AU)−2.5 g cc−1 . These
values are only meant to be illustrative, and variations are
possible without changing the basic properties discussed here.
Figure 1 shows the Mg pressure profile of the resulting disk
for four different volume mixing ratios of a hypothetical gas,
assuming all Mg is in vapor. The 1× curve is based on the
solar mixing ratio of Mg, an ideal gas, and a mean molecular
weight μ = 2.3. Curves represented by 10×, etc., are 10 ×
the Mg partial pressure at solar abundance (μ will be different
in these cases). For a solar abundance nebula, if all the Mg
were in vapor, the partial pressure would become much less
than the saturation pressure for temperatures above ∼1250 K.
J ≈ J0 exp (−E0 /(RT ))(PH2 /P0 )1/2 .
The local mass density of solids can become very large
in the inner regions of disks, even with a fixed solid-to-gas
ratio. This will promote an increased solid collision rate, which
could be further enhanced if any solid concentrations were to
occur (e.g., some midplane settling). Nonetheless, solids are
expected to evaporate (either from liquidus or solidus) at the
high temperatures of the inner disk, causing many solids to be
lost in this high-density region. The exact boundary depends on
several factors, but it is often assumed that solids will evaporate
at temperatures in excess of 1400–1500 K. To determine whether
particles are lost to evaporation, it is necessary to take into
account the saturation vapor pressure of rock and the timescale
for the evaporation of solid material relative to collision times
as applicable to conditions in disks at radii ∼0.1 AU.
Whenever the partial pressure of a vapor is equal to its
saturation pressure, evaporation and re-condensation will be in
equilibrium. For a gas with only a single species, the evaporation
rate (mole per area per time) and saturated vapor pressure are
related by the Hertz–Knudsen relation:
J =
γ Ps
,
(2π mRT )1/2
(1)
where m is the weight of the species, T is the gas temperature
with gas constant R, γ is the evaporation coefficient, and Ps is
the saturation vapor pressure (e.g., Richter et al. 2002). As a
proxy for evaporation of rock, we focus on Mg in the context
5 We use a protosolar value A = 7.62, and assume a hydrogen number
density 0.92 the total number density.
2
(2)
The Astrophysical Journal Letters, 792:L27 (4pp), 2014 September 10
Boley, Morris, & Ford
For an enhancement of Mg vapor that is 10 times the solar
abundance (10×), evaporation is suppressed until about 1470 K.
At even higher concentrations of 30 Z and 100 Z, evaporation
will be suppressed until temperatures of 1600 and 1800 K are
reached, respectively. For high volume mixing ratios, which
may be obtained through high initial metal abundances and/or
inward drift of solids, net evaporation can be suppressed.
The above limit assumes that newly formed vapor can diffuse
away from an evaporating solid instantaneously. As pointed
out by Cuzzi & Alexander (2006) the overlapping effects of
evaporating solids can allow ensembles of solids/droplets to
equilibrate with their own collective vapor. While their work
was done in the context of chondrule formation (unlike this
study), the basic physics is general and applicable to our model
for the rapid growth of solids in the inner nebula. Net evaporation
can be suppressed whenever exp(−nπ s 2 γ vth t) = exp(−ξ ) ∼ 0
(see Cuzzi & Alexander 2006), where s is some representative
size for solids that have a number density n, v th is the thermal
velocity of the vapor, and γ is the evaporation coefficient. The
time t is a characteristic time for the problem, which we take to
be the nominal evaporation timescale:
t ev ≈ 2 ρm s/(3J 140 g mole−1 ).
of the concentration of solids in the midplane due to settling
(see K defined next section). Even for very high midplane
concentrations, self-shielding effects play a role in limiting
the net evaporation rate of solids. We also need to consider
whether the solids themselves can move out of their own vapor
cloud due to radial drift. Assuming that centimeter-sized solids
migrate inward between about vdrift ∼ 10 and 100 cm s−1 ,
the radial dimension of the evaporation front can extend for
at least ∼vdrift t ev ∼ 100–2000 km. However, inward moving
solids will always produce a collective vapor trail that will be
seen by solids entering the evaporation front, and diffusion in
the vertical direction may still be the most limiting condition.
3. COLLISIONS IN THE INNER REGIONS OF
PLANET-FORMING DISKS
The calculations presented here are based on the evaporation
of Mg. However, Si is expected to have a similar behavior
(Richter et al. 2002). While the detailed rates will change when
considering bulk compositions of rocks, the overall picture
should remain valid: rocks will not necessarily be destroyed
by evaporation in the inner nebula. Moreover, collisions will
be very frequent in this environment, and the solids may be
partially molten (discussed more below).
The mass growth rate for a solid of size s colliding with
solids of similar or smaller sizes is ṁ = ρs π s 2 vvel for local
solid mass density ρs , cross section π s 2 , and relative velocity
v rel . The mass growth can be related to size growth by ṡ =
(ρs /ρm )(vvel /4) for particle internal density ρm . Both the solid
mass density and the relative velocity will depend on the
solid size and the degree of turbulence in the disk, which can
be described using the α formalism (Cuzzi & Hogan
2003).
√
Following (Dubrulle et al. 1995), vrel = α 1/2 c( 2τ /(1 + τ ))
for sound speed c (∼3 km s−1 at 0.1 AU). The amount of
settling, and hence midplane concentration, K = (τ/α)1/2 (1 +
(α/τ ))1/2 , where √
ρs = Kρ0,s . Combining these relations,
ṡ = (ρ0,s /ρm )(c/2 2)(τ/(1 + τ ))(1+(α/τ ))1/2 . When τ is large,
the growth approaches the Safronov limit. At small τ , radial drift
by larger solids and Brownian motion will prevent the growth
rate from becoming trivially small. Note that the growth rate
becomes nearly independent of α, except at large α/τ .
As done for the ρs calculation above, we take ρ0,s ∼
10−9 g cc−1 for our envisaged disk. At these distances, sizes
s ∼ 10 cm have τ ∼ 1 (based on Weidenschilling 1977, but for
conditions at 0.1 AU). As particles approach this size, they will
grow from collisions at about 2 cm day−1 , which will increase
to about 4 cm day−1 as τ becomes large. We can thus expect
20–40 cm of growth per orbit. The peak radial drift timescale
is of the order of 100 local orbits, allowing the solid to grow
above τ ∼ 100. The actual radial drift rate will decay rapidly
during this time, creating a stable and limiting situation for
growth. Higher metallicities or local concentrations of solids
will enhance this effect.
These results are in reasonable agreement with existing
literature. Birnstiel et al. (2010) show that the meter barrier
can be overcome for perfect sticking and a drift rate efficiency
of 0.75 for growth at ∼ 0.2 AU in their model. The growth
rate per orbit will be more than twice as fast at 0.1 AU as
it is at 0.2 AU for reasonable disk models, and no reduction
of drift is necessary. Moreover, as discussed next, our results
physically motivate the assumption of perfect or near-perfect
sticking, which the authors assumed as a test case.
(3)
The factor of two takes into account that two Mg must be lost for
every forsterite. A 1 μm grain with ρm = 2.5 g cc−1 at 1500 K
will evaporate in about 2 minutes at PH2 ∼ 10 mbar, where this
H2 pressure is meant to be illustrative of the types of conditions
that can be found at 0.1 AU around a solar-mass star. In contrast,
a centimeter-sized grain requires 10–20 days to evaporate under
these conditions, or ∼1–2 orbital periods at 0.1 AU.
Now consider the case where all the mass is in solids
of size s. In this case, ξ = γ (3ρs /4ρm )(vth /s)t ev . To allow
evaporation and re-condensation to equilibrate, ξ 3–6, which
corresponds to ρs 1–2 × 10−8 g cc−1 (γ ∼ 0.007 at 1500 K).
However, equilibration is not strictly necessary, as we only
require the effective evaporation time to be much longer than the
coagulation timescale. For ξ ∼ 1, the effective evaporation time
will be increased by a factor of few over the nominal tev , with a
corresponding ρs 2–3 × 10−9 g cc−1 . A reasonable estimate
of the total mass fraction of refractory material that is drifting
into radii ∼ 0.1 AU is approximately 0.00375 (i.e., for
solar metallicity and no additional concentration). Accordingly,
in our envisaged disk ρs ∼ 10−9 (0.1)−2.5 0.00375 g cc−1 ∼
10−9 g cc−1 . Even if the far-field partial pressure does not
exceed the saturation pressure at ∼0.1 AU, inward drifting
1 cm solids will begin to show notable self-shielding effects
at concentrations of a few times solar, and suppression of net
evaporation could occur for concentrations 10–20 solar. This
estimate does not strongly depend on the actual representative
grain size due to the direct dependence of tev on s.
The above situation √
becomes sustainable if the evaporation
front is a few times ∼ Dt ev , i.e., the distance vapor diffuses
away from the ensemble of solids during evaporation. The
diffusion coefficient D ∼ (0.00014/P (bar))T (K)1.5 cm2 s−1
sets the rate at which vapor can diffuse through H2 gas. For
our envisaged conditions, D ∼ 800 cm2 s−1 , requiring the
evaporation front to be L 1 km to mitigate the effects
of diffusion. The gas scale height H at 0.1 AU is about
400,000 km in our envisaged disk. Thus, the evaporation front
only needs to extend over a small fraction of H. Solids will
evaporate over a vertical distance that is comparable to the solid
vertical scale height Hs , which we take to be comparable to
the overall size of L. The ratio Hs /H is roughly the inverse
3
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Boley, Morris, & Ford
in planet-forming disks and can lead to in situ formation
of planets at short orbital periods. The mechanism has two
components, both resulting from the environment of the inner
nebula. (1) Net evaporation from solid surfaces can become
suppressed by high partial pressures, partly due to self-shielding
effects. (2) Collisional growth rates will be very large at
short orbital periods as long as collisional destruction can be
mitigated, which we suggest can be facilitated by collisions
between partially molten solids. In this context, solids migrate
to small disk radii, experience some melting without complete
evaporation, and grow beyond the fast radial drift regime. This
mechanism does not require additional instabilities or radially
localized total pressure enhancements, although such conditions
would aid this process.
While the above processes may be very common, additional
work is required to understand the diversity of planetary system
architectures. For example, the fraction of stars initially hosting STIP-like systems may be greater than that observed for
main-sequence stars due to subsequent orbital evolution sculpting the population of long-term stable planetary systems observed by Kepler. It is also possible that disk processes lead to
bifurcations of planetary system architectures early in the formation process, with subsequent evolution playing a moderate
role.
The assumption that collisions lead to growth is reasonable
if kinetic energy can be dissipated during collisional growth.
If solids are partially molten, then their viscosity may provide
this dissipation (below). Ebel & Grossman (2000) showed that
silicate melts will be stable for a range of high temperatures
(including 1500 K) at total pressures of 1 mbar and at rock vapor
enhancements of about 100 solar. Fractional melts of say 10%
will require less extreme conditions. These experiments were
only conducted at 1 mbar, while 10 mbar is more representative
of the pressure in the inner nebula. The stability of melts is
dependent on the pressure, which will reduce the necessary
vapor enhancement further. For these reasons, the enhancement
of 100 should be taken as an upper limit, but such a value may
be attainable.
Ciesla (2006), motivated by compound chondrules, showed
that hard spheres with s ∼ 0.3 mm and a thin viscous surface
layer can survive collisions ∼100 m s−1 (viscosity η ∼ 100
poise). At larger speeds or larger solids, the thin surface
layer will not dissipate all of the kinetic energy without a
comparable increase in viscosity. In the model explored by
Ciesla, failure to dissipate kinetic energy did not by itself
mean the solids were destroyed, rather, the model could no
longer predict the outcome. However, a free-floating rock or
boulder that is just experiencing melt may be better described
as a solid suspension. If the suspension is characterized by
densely packed particles surrounded by layers of melted rock,
then the rheology can be non-Newtonian and the effective
viscosity can become many orders of magnitude larger than
the fluid’s viscosity in isolation (e.g., Stickel & Powell 2005).
For example, if the particles in the suspension are hard spheres
with a volume filling ratio of φ, with some maximum possible
volume ratio φm ∼ 0.6, then the effective viscosity η = η(1 +
(5φ/(1 − φ/φm )))2 as derived experimentally. Large viscosities
could dissipate significant kinetic energy, including collisions
in excess of 100 m s−1 . A rough estimate of this dissipation is
Eη ∼ 4π η vs 2 /3. Comparing this with the kinetic energy gives
Eη /Ek ∼ 2η/(ρm sv). For a rock with s ∼ 100 cm to survive a
collision at v ∼ 100 m s−1 with a comparable impactor requires
η ∼ 1 to 2 Mpoise. This is a factor of 104 larger than what
may be typical for molten material alone, but is feasible for
a suspension where φ → φm . At small φ, collisions of large
rocks will not necessarily lead to growth, but because collisions
should be taking place continuously, growth may begin at the
onset of the initial melt stages.
The growth from planetesimals to planets is harder to estimate. In particular, if the typical relative velocity for collisions
becomes too high, the effective viscosities required to dissipate
sufficient collisional energy may no longer be attainable. These
difficulties may be overcome, at least in part, by the highly dissipative and high-density environment of the inner nebula, but
this remains a topic for further study and will be needed to test
the full viability of this model.
The authors thank Conel Alexander and Fred Ciesla for
their invaluable comments. We also thank the referees for their
helpful comments and suggestions. A.C.B.’s contribution was
supported by The University of British Columbia. M.A.M.
thanks Erik Asphaug and the Center for Meteorite Studies
at Arizona State University for support. E.B.F. acknowledges
support from the Penn State Center for Exoplanets and Habitable
Worlds and a NASA Kepler Participating Scientist Program
award (NNX12AF73G).
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4. DISCUSSION
We have identified one potential solution to the outstanding
meter barrier problem. If correct, the process should be common
4